I have already sent my scores and transcripts to the schools and I haven't even started the online applications for most of them. Would it be a problem that my scores and transcripts arrive before my applications? Would the admission people have trouble matching my stuff?

At one point during my last year's applications, I had asked an admissions person this question. Basically if they get anything from you, they start a file for you, even if they don't have your application yet.

Related: At one school they actually had 2 files for me, but only 1 application. The two files were because the oh-so-wonderful ETS screwed up my birthday and took 6 weeks to fix it. So the school had a file with my name and all my information, but my GRE scores were stuck in that file and nothing else. So at least for some programs it is most likely not a problem.